Formulas for risk of failure

The concept of stress strength inference plays an important role in modeling problems with respect to engineering fracture mechanics, see, e.g. [1–21]. The risk of failure of a system can be expressed by the probability R = Pr(X2 < X1), where X1 and X2 are some random variables representing the stress and the strength experienced by the system. In this note, we derive a comprehensive collection of formulas for R by assuming the most commonly known models for X1 and X2. We feel that this work could serve as a useful reference for engineering fracture mechanics.